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\title{复变函数第二章：解析函数}
\author{ZYQ ET AL}
\date{2024年3月18日}

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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

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\begin{frame}{第二章目录 }

\begin{enumerate}

\item[2.1.] 解析函数的概念与柯西-黎曼方程
\item[2.2.] 初等解析函数
\item[2.3.] 初等多值函数
\item[2.4.] 部分习题

\end{enumerate}

\end{frame}

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\begin{frame}{1.1. 导数与微分的概念 }

\begin{itemize}

\item  {\color{red}问题： 
设函数 $w=f(z)$ 在包含 $z_0$ 的区域 $D$ 内有定义。
\begin{enumerate}%[label={(\arabic*)}]
\item  {\color{red}什么时候称函数 $f(z)$ 在点 $z_0$ 可导？}
\item  {\color{red}什么时候称函数 $f(z)$ 在点 $z_0$ 可微？}
\item  {\color{red}什么时候称函数 $f(z)$ 在点 $z_0$ 解析？}
%什么是 $w=f(z)$ 在点 $z_0$ 的微分？
\end{enumerate} 

}

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2. 例子2.1.  }

\begin{itemize}

\item  {\color{red}问题：证明函数 $f(z)=\overline{z}$ 在复数平面上处处不可微。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.3. 例子2.2.  }

\begin{itemize}

\item  {\color{red}问题：证明函数 $f(z)=z^n$ 在复数平面上处处可微，并求导数。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4. 例子2.3-2.4-2.5 }

\begin{itemize}

\item  {\color{red}问题：什么时候称函数 $w=f(z)$ 为区域 $D$ 内的解析函数？
验证下述函数在有定义的区域中是解析函数：
\begin{eqnarray*}
P(z) &=& a_0z^n + a_1z^{n-1} + \cdots + a_n, \\ 
\frac{P(z)}{Q(z)}&=& \frac{a_0z^n + a_1z^{n-1} + \cdots + a_n}{b_0z^m + b_1z^{m-1} + \cdots + b_m}, \\ 
f(z) &=& (3z^2-4z+5)^{11}. 
\end{eqnarray*}
}

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.5. 定理2.1. 可微的必要条件 }

\begin{itemize}

\item  {\color{red}问题：设函数 $f(z)=u(x,y)+iv(x,y)$ 在区域 $D$ 内有定义，
且在 $D$ 内一点 $x+iy$ 可微，则必有  }

\begin{enumerate}%[label={(\arabic*)}]

\item  {\color{red}偏导数 $u_x, u_y, v_x, v_y$ 在点 $(x,y)$ 存在。}

\item  {\color{red}$u(x,y), v(x,y)$ 在点 $(x,y)$ 满足柯西-黎曼方程：

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}. $$ } 

\end{enumerate}

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.6. 例子2.6. }

\begin{itemize}

\item  {\color{red}问题：证明函数 $f(z)=\sqrt{|xy|}$ 在 $z=0$ 满足定理2.1的必要条件，但是在 $z=0$ 不可微。
}


\end{itemize}

\end{frame}

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\begin{frame}{1.7. 定理2.2. 可微的充要条件 }

\begin{itemize}

\item  {\color{red}问题：设函数 $f(z)=u(x,y)+iv(x,y)$ 在区域 $D$ 内有定义，则 $f(z)$ 在 $D$ 内一点 $x+iy$ 可微的充要条件是  }

\begin{enumerate}%[label={(\arabic*)}]

\item  {\color{red}二元函数 $u(x,y), v(x,y)$ 在点 $(x,y)$ 可微。}

\item  {\color{red}$u(x,y), v(x,y)$ 在点 $(x,y)$ 在点 $(x,y)$ 满足柯西-黎曼方程：

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}. $$ } 

\end{enumerate} 

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.8. 推论2.3. 可微的充分条件 }

\begin{itemize}

\item  {\color{red}问题： 设函数 $f(z)=u(x,y)+iv(x,y)$ 在区域 $D$ 内有定义，则 $f(z)$ 在 $D$ 内一点 $x+iy$ 可微的充分条件是  }

\begin{enumerate}%[label={(\arabic*)}]

\item  {\color{red}偏导数 $u_x, u_y, v_x, v_y$ 在点 $(x,y)$ 连续。}

\item  {\color{red}$u(x,y), v(x,y)$ 在点 $(x,y)$ 在点 $(x,y)$ 满足柯西-黎曼方程：

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}. $$ } 

\end{enumerate} 
\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.9. 例子2.7. }

\begin{itemize}

\item  {\color{red}问题：讨论函数 $f(z)=|z|^2$ 的解析性。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.10. 例子2.8. }

\begin{itemize}

\item  {\color{red}问题：讨论函数 $f(z)=x^2-iy$ 的可微性和解析性。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.11. 例子2.9. }

\begin{itemize}

\item  {\color{red}问题：设 $f(z)=x^2+axy+by^2+i(cx^2+dxy+y^2)$, 问常数 $a,b,c,d$ 取何值时，$f(z)$ 在复平面内处处解析？ }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.12. 例子2.10. }

\begin{itemize}

\item  {\color{red}问题：证明函数 $f(z)=e^x(\cos y + i\sin y)$ 在复平面上解析，且 $$f\,'(z)=f(z). $$ }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.13. 例子2.11. }

\begin{itemize}

\item  {\color{red}问题：设 $f(z)=u(x,y)+iv(x,y)$ 在区域 $D$ 内解析，并且 $v=u^2$, 求 $f(z)$.  }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.14. 例子2.12. }

\begin{itemize}

\item  {\color{red}问题：设函数 $f(z)=u(x,y)+iv(x,y)$ 在区域 $D$ 内解析，且 $f\, '(z)\neq 0 \, (z\in D)$, 证明 
$u(x,y)=c_1, v(x,y)=c_2$ （$c_1,c_2$ 为常数），是 $D$ 内两组正交曲线族。
}

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.15. 定理2.6. }

\begin{itemize}

\item  {\color{red}问题：将复数 $z$ 表示成指数形式 $z=re^{i\theta}$, 设函数 $w=f(z)$ 表示成 $w=u(r,\theta)+iv(r,\theta$, 
若 $u(r,\theta), v(r,\theta)$ 可微，且 $$u_r = \frac{1}{r}v_\theta, u_\theta=-rv_r,$$
则 $f(z)=f(r,\theta)$ 可微。
 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.16. 定理2.7. }

\begin{itemize}

\item  {\color{red}问题：设 $u(x,y)$ 和 $v(x,y)$ 在区域 $D$ 内有一阶连续偏导数，则 $f(z)=u(x,y)+iv(x,y)$ 在 $D$ 内解析的充分必要条件为 $$\frac{\partial f}{\partial \bar{z}}(z) =0. $$ }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{2.1.  }

\begin{itemize}

\item  {\color{red}问题：写出指数函数 $e^z$ 的定义，证明基本性质。  }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{2.2. 例子2.13. }

\begin{itemize}

\item  {\color{red}问题：求 $e^z$ 的基本周期。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{2.3.  }

\begin{itemize}

\item  {\color{red}问题：写出正弦函数和余弦函数的定义。证明基本性质。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{2.4. 例子2.14.  }

\begin{itemize}

\item  {\color{red}问题：求 $\sin(1+2i)$ 的值。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{2.5. 例子2.15. }

\begin{itemize}

\item  {\color{red}问题：求正弦函数的基本周期。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{2.6.  }

\begin{itemize}

\item  {\color{red}问题：写出正切函数和余切函数的定义。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{2.7.  }

\begin{itemize}

\item  {\color{red}问题：写出双曲正弦函数、双曲余弦函数、双曲正切函数、双曲余切函数的定义。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.1. 初等多值函数 }

\begin{itemize}

\item  {\color{red}问题：什么时候称函数 $f(z)$ 在区域 $D$ 内是单叶的？ }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.2. 例子2.17. }

\begin{itemize}

\item  {\color{red}问题：对图中的三条具有相同起点和终点的简单曲线，计算 $$\Delta_L \mathrm{Arg}(z). $$

 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.3. 定义2.9.  }

\begin{itemize}

\item  {\color{red}问题：写出根式函数 $w=\sqrt[n]{z}$ 的定义。 }
\begin{enumerate}
\item  {\color{red}确定根式函数的单叶性区域。}
\item  {\color{red}分出根式函数的单值解析分支。}
\item  {\color{red}什么是根式函数的支点和支割线？}
\item  {\color{red}什么是根式函数的主值支？}
\end{enumerate}

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.4. 例子2.18. }

\begin{itemize}

\item  {\color{red}问题：设 $w=\sqrt[3]{z}$ 确定在 $\mathbb{C}-(-\infty,0]$ 上，并且 $w(i)=-i$, 试求 $w(-i)$ 的值。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.5.  }

\begin{itemize}

\item  {\color{red}问题：写出对数函数的定义。 }
\begin{enumerate}
\item  {\color{red}确定对数函数的单叶性区域。}
\item  {\color{red}分出对数函数的单值解析分支。}
\item  {\color{red}什么是对数函数的支点和支割线？}
\item  {\color{red}什么是根式函数的主值支？}
\end{enumerate}

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.6. 例子2.19. }

\begin{itemize}

\item  {\color{red}问题：设 $a>0$, 则 $$\mathrm{Ln}\, a = \ln a + 2k\pi i \,\,\, (k=0,\pm 1, \pm 2, \cdots), $$  
其主值就是通常的实对数 $\ln a$. 
}

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.7. 例子2.20. }

\begin{itemize}

\item  {\color{red}问题：计算 $\mathrm{Ln}\, i$.  }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.8. 例子2.21.  }

\begin{itemize}

\item  {\color{red}问题：计算 $\mathrm{Ln}\, (3+4i)$.  }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.9. 定义2.11.  }

\begin{itemize}

\item  {\color{red}问题：写出一般的幂函数 $w=z^a$ 的定义。  }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.10. 例子2.22-2.23. }

\begin{itemize}

\item  {\color{red}问题：求 $i^i$ 和 $2^{1+i}$. }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.11. 例子2.24. }

\begin{itemize}

\item  {\color{red}问题：求下述函数的支点：
\begin{eqnarray*}
f(z) &=& \sqrt{z(1-z)}, \\ 
f(z) &=& \sqrt[3]{z(1-z)}, \\ 
f(z) &=& \sqrt{z(z-1)(z-2)(z-3)(z-4)}, \\ 
f(z) &=& \sqrt[3]{z(z-1)(z-2)(z-3)(z-4)}. 
\end{eqnarray*}

 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.12. 例子2.25.  }

\begin{itemize}

\item  {\color{red}问题：证明 $f(z)=\sqrt[3]{z(1-z)}$ 在将 $z$ 平面适当割开后，能分出三个单值解析分支，并求出在 $z=2$ 取负值的那个分支在 $z=i$ 的值。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.13. 例子2.26. }

\begin{itemize}

\item  {\color{red}问题：证明函数 $\mathrm{Ln}\,(1-z^2)$ 在割去从 $-1$ 到 $i$ 的直线段和从 $i$ 到1的直线段、与射线 $x=0$且 $y\ge 1$ 的 $z$ 平面内能分出单值解析分支。并求 $z=0$ 时等于零的那一支在 $z=2$ 的值。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.14.  }

\begin{itemize}

\item  {\color{red}问题：写出反正切函数、反正弦函数、反余弦函数的定义。用对数函数表示这些反三角函数。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.15.  }

\begin{itemize}

\item  {\color{red}问题：用对数函数表示双曲函数的反函数。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.16. 例子2.27-2.28. }

\begin{itemize}

\item  {\color{red}问题：求 $\mathrm{Arcsin}\, 2$ 和 $\mathrm{Arctan}\, (2i)$.  }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.1. 习题3 }

\begin{itemize}

\item  {\color{red}问题：设 
$$f(z) = \left\{ \begin{array}{ll}
\frac{x^3-y^3+i(x^3+y^3)}{x^2+y^2}, & z=x+iy\neq 0, \\
0, & z=0,
\end{array}\right. $$
证明 $f(z)$ 在原点满足柯西-黎曼方程，但是不可微。
 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.2. 习题4 }

\begin{itemize}

\item  {\color{red}问题：证明下述函数在 $z$ 平面上任何点都不解析：
$$
(1) |z|;\,\, 
(2) x+y; \,\, 
(3) \mathrm{Re}(z); \,\, 
(4) \frac{1}{\bar{z}}. 
$$
}

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.3. 习题5 }

\begin{itemize}

\item  {\color{red}问题：判断下列函数的可微性与解析性：
\begin{eqnarray*}
(1) && f(z)=xy^2+ix^2y; \\
(2) && f(z)=x^2+iy^2; \\
(3) && f(z)=2x^3+3iy^3; \\
(4) && f(z)=x^3-3xy^2+i(3x^2y-y^3). 
\end{eqnarray*}

 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.4. 习题8 }

\begin{itemize}

\item  {\color{red}问题：证明下列函数在复平面上解析，并求其导数： 
$$f(z) = x^3+3x^2yi-3xy^2-y^3i. $$
}

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.5. 习题10 }

\begin{itemize}

\item  {\color{red}问题：设 $z=x+iy$, 求 
$$
(1)\,\, |\exp(i-2z)|; \,\,
(2)\,\, |\exp(z^2)|; \,\,
(3)\,\, \mathrm{Re}(\exp(1/z)). 
$$
 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.6. 习题11 }

\begin{itemize}

\item  {\color{red}问题：求下列值及其主值：
$$
(1)\,\, \mathrm{Ln}\, (3-\sqrt{3}i) ; \,\,
(2)\,\, (2i)^i. 
$$

 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.7. 习题14 }

\begin{itemize}

\item  {\color{red}问题：证明：
$$
(1)\,\, \lim\limits_{z\to 0} \frac{\sin z}{z}=1; \,\,\,\,
(2)\,\, \lim\limits_{z\to 0} \frac{\exp(z)-1}{z}=1; \,\,\,\,
(3)\,\, \lim\limits_{z\to 0} \frac{z-z\cos z}{z-\sin z}=3. 
$$
 
 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.8. 习题20 }

\begin{itemize}

\item  {\color{red}问题：解方程：
\begin{eqnarray*}
(1)&& \exp(z)=1+\sqrt{3}i; \\ 
(2)&& \ln z = \pi i/2; \\ 
(3)&& 1+\exp(z)=0; \\ 
(4)&& \cos(z)+\sin(z)=0; \\ 
(5)&& \tan(z)=1+2i. 
\end{eqnarray*}
 
 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.9. 习题21 }

\begin{itemize}

\item  {\color{red}问题：设 $z=re^{i\theta}$, 证明 $$\mathrm{Re}[\ln(z-1)] = \frac{1}{2} \ln (1+r^2-2r\cos\theta). $$ }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.10. 习题22 }

\begin{itemize}

\item  {\color{red}问题：设 $w=\sqrt[3]{z}$ 确定在 $\mathbb{C}-[0,\infty)$ 上，并且 $w(i)=-i$, 求 $w(-i)$.  }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.11. 习题24 }

\begin{itemize}

\item  {\color{red}问题：求 $(1+i)^i$ 与 $3^i$ 的值。 }

\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{4.12. 习题H2 }

\begin{itemize}

\item  {\color{red}问题：设 $f(z)=\frac{z}{1-z}$, 证明当 $|z|<1$ 时，有
$$\mathrm{Re} \left[ 1+ z\frac{f\,''(z)}{f\,'(z)} \right] >0. $$
}

\item  解答：


\end{itemize}

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\begin{frame}{4.13. 习题H4 }

\begin{itemize}

\item  {\color{red}问题：设 $f(z)=u+iv\in C^1(D)$, 证明：
$$ \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}  = \left\lvert \frac{\partial f}{\partial z}\right\rvert^2 
-\left\lvert \frac{\partial f}{\partial \bar{z}}\right\rvert^2. $$

}

\item  解答：


\end{itemize}

\end{frame}


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